Coburn J.W. Algebra and Trigonometry
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200 CHAPTER 2 Relations, Functions, and Graphs 2-50
b. From the graph, the secant line for 1992 to 1996 appears to have a greater slope.
1986–1992 CDs 1992–1996 CDs
For 1986 to 1992: ; for 1992 to 1996: , a growth rate much
higher than the earlier period.
Now try Exercises 102 and 103
m 92.75m 59.2
92.75 59.16
371
4
355
6
¢y
¢x
779 408
16 12
¢y
¢x
408 53
12 6
1992: 112, 4082, 1996: 116, 77921986: 16, 532, 1992: 112, 4082
2.4 EXERCISES
CONCEPTS AND VOCABULARY
Fill in each blank with the appropriate word or phrase.
Carefully reread the section if needed.
1. If a relation is given in ordered pair form, we state
the domain by listing all of the
coordinates in a set.
2. A relation is a function if each element of the
is paired with element
of the range.
3. The set of output values for a function is called the
of the function.
4. Write using function notation: The function f
evaluated at 3 is negative 5:
5. Discuss/Explain why the relation is a
function, while the relation is not. Justify
your response using graphs, ordered pairs, and
so on.
6. Discuss/Explain the process of finding the domain
and range of a function given its graph, using
vertical and horizontal boundary lines. Include a
few illustrative examples.
x y
2
y x
2
DEVELOPING YOUR SKILLS
Determine whether the mappings shown represent
functions or nonfunctions. If a nonfunction, explain how
the definition of a function is violated.
7.
Indira Gandhi
Clara Barton
Margaret Thatcher
Maria Montessori
Susan B. Anthony
Woman Country
Britain
U.S.
Italy
India
8.
Hawaii
Roots
Shogun
20,000 Leagues
Under the Sea
Where the Red
Fern Grows
Book Author
Rawls
Verne
Haley
Clavell
Michener
D. You’ve just learned how
to apply the rate-of-change
concept to nonlinear
functions
College Algebra—
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9.
10.
Determine whether the relations indicated represent
functions or nonfunctions. If the relation is a nonfunction,
explain how the definition of a function is violated.
11. ( , 0), (1, 4), (2, ), (4, 2), ( , 6), (3, 6), (0, ),
(4, ), and (6, 1)
12. ( ), ( , 3), (4, 0), ( ), (1, ),
(0, 9), (2, ), (3, ), and ( , 7)
13. (9, ), ( , 6), (6, ), (4, ), (2, ),
(1, 8), (0, ), ( ), and ( , 4)
14. (1, ), ( , 64), ( , 49), (5, ), ( , 25),
(13, ), ( , 9), (34, ), and ( , 1)
15. 16.
17. 18.
Determine whether or not the relations given represent
a function. If not, explain how the definition of a
function is violated.
19. 20.
x
y
5
5
5
5
x
y
55
5
5
x
y
55
5
5
(3, 4)
(3, 2)
(3, 3)
(1, 1)
(5, 2)
(1, 4)
x
y
55
5
5
(2, 3)
(5, 1)
(4, 5)
(3, 4)
(1, 2)
(2, 4)
x
y
55
5
5
(3, 5)
(5, 3)
(0, 2)
(3, 4)
(1, 3)
(4, 2)
x
y
5
5
5
5
(3, 4)
(5, 0)
(1, 1)
(1, 4)
(2, 4)
(4, 2)
5542116
8363281
62, 72
2110710
528
63, 557, 5
5
1553
Country Language
Canada
Japan
Brazil
Tahiti
Ecuador
Japanese
Spanish
French
Portuguese
English
MJ
The Mailman
The Doctor
The Iceman
The Shaq
Basketball star Reported height
7'1"
6'6"
6'7"
6'9"
7'2"
21. 22.
23. 24.
25. 26.
27. 28.
29. 30.
Graph each relation using a table, then use the vertical
line test to determine if the relation is a function.
31. 32.
33. 34.
Determine whether or not the relations indicated
represent a function, then determine the domain and
range of each.
35. 36.
x
y
55
5
5
x
y
55
5
5
x
y 2
y 1x 22
2
y 2
3
xy x
x
y
55
5
5
x
y
55
5
5
x
y
55
5
5
x
y
5
5
5
5
x
y
55
5
5
x
55
5
y
5
x
y
55
5
5
x
y
55
5
5
x
y
55
5
5
x
y
55
5
5
2-51 Section 2.4 Functions, Function Notation, and the Graph of a Function 201
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37. 38.
39. 40.
41. 42.
43. 44.
45. 46.
Determine the domain of the following functions.
47. 48.
49. 50.
51. 52.
53. 54.
55. 56.
57. 58.
s t
2
3t 10m n
2
3n 10
y
11
19
x 89y
17
25
x 123
p
q 7
q
2
12
u
v 5
v
2
18
w1x2
x 4
x
2
49
v1x2
x 2
x
2
25
p1a2 15a 2
h1a2 13a 5
g1x2
2
3 x
f1x2
3
x 5
x
y
55
5
5
x
y
55
5
5
x
y
55
5
5
x
y
5
5
5
5
x
y
55
5
5
x
y
55
5
5
x
y
55
5
5
x
y
55
5
5
x
y
55
5
5
x
y
55
5
5
59. 60.
61. 62.
63. 64.
65. 66.
67. 68.
Determine the value of and f(c 1),
then simplify as much as possible.
69. 70.
71. 72.
Determine the value of h(3), , h(3a), and h(a 2),
then simplify as much as possible.
73. 74.
75. 76.
Determine the value of g(4), g(2c), and g(c 3),
then simplify as much as possible.
77. 78.
79. 80.
Determine the value of p(5), p(3a), and p(a 1),
then simplify as much as possible.
81. 82.
83. 84.
Use the graph of each function given to (a) state the
domain, (b) state the range, (c) evaluate f(2), and
(d) find the value(s) x for which (k a constant).
Assume all results are integer-valued.
85. 86.
x
y
55
5
5
x
y
55
5
5
k 3k 4
f1x2 k
p1x2
2x
2
3
x
2
p1x2
3x
2
5
x
2
p1x2 14x 1p1x2 12x 3
p(
3
2
),
g1r2 r
2
hg1r2 r
2
g1r2 2rhg1r2 2r
g(
3
2
),
h1x2
4
x
x
h1x2
5
x
x
h1x2
2
x
2
h1x2
3
x
h(
2
3
)
f 1x2 2x
2
3xf1x2 3x
2
4x
f 1x2
2
3
x 5f 1x2
1
2
x 3
f(6), f(
3
2
), f(2c),
p1x2
7
15 x
h1x2
2
14 x
g1x2
A
4
3 x
f 1x2
A
5
x 2
y
1x 1
3x 2
y
1x 2
2x 5
y
2
x 4
x
2
2x 15
y
1
x
x
2
3x 10
y
x 2
3y 2
x
1
202 CHAPTER 2 Relations, Functions, and Graphs 2-52
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87. 88.
55
5
x
y
5
55
5
5
y
x
k 3k 1 89. 90.
55
5
5
x
y
x
y
55
5
5
k 1k 2
2-53 Section 2.4 Functions, Function Notation, and the Graph of a Function 203
WORKING WITH FORMULAS
91. Ideal weight for males:
The ideal weight for an adult male can be modeled
by the function shown, where W is his weight in
pounds and H is his height in inches. (a) Find the
ideal weight for a male who is 75 in. tall. (b) If I
am 72 in. tall and weigh 210 lb, how much weight
should I lose?
92. Celsius to Fahrenheit conversions:
The relationship between Fahrenheit degrees and
degrees Celsius is modeled by the function shown.
(a) What is the Celsius temperature if ?
(b) Use the formula to solve for F in terms of C, then
substitute the result from part (a). What do you
notice?
°F 41
C
5
9
(F 32)
W(H)
9
2
H 151
93. Pick’s theorem:
Picks theorem is an interesting yet little known
formula for computing the area of a polygon drawn
in the Cartesian coordinate system. The formula can
be applied as long as the vertices of the polygon are
lattice points (both x and y are integers). If B
represents the number of lattice points lying directly
on the boundary of the polygon (including the
vertices), and I represents the number of points in
the interior, the area of the polygon is given by the
formula shown. Use some graph paper to carefully
draw a triangle with vertices at (3, 1), (3, 9), and
(7, 6), then use Pick’s theorem to compute the
triangle’s area.
A
1
2
B I 1
APPLICATIONS
94. Gas mileage: John’s old ’87 LeBaron has a 15-gal
gas tank and gets 23 mpg. The number of miles he
can drive is a function of how much gas is in the
tank. (a) Write this relationship in equation form
and (b) determine the domain and range of the
function in this context.
95. Gas mileage: Jackie has a gas-powered model boat
with a 5-oz gas tank. The boat will run for 2.5 min
on each ounce. The number of minutes she can
operate the boat is a function of how much gas is in
the tank. (a) Write this relationship in equation
form and (b) determine the domain and range of
the function in this context.
96. Volume of a cube: The volume of a cube depends
on the length of the sides. In other words, volume
is a function of the sides: . (a) In practical
terms, what is the domain of this function?
(b) Evaluate V(6.25) and (c) evaluate the function
for
97. Volume of a cylinder: For a fixed radius of 10 cm,
the volume of a cylinder depends on its height. In
other words, volume is a function of height:
s 2x
2
.
V1s2 s
3
. (a) In practical terms, what is the
domain of this function? (b) Evaluate V(7.5) and
(c) evaluate the function for .
98. Rental charges: Temporary Transportation Inc.
rents cars (local rentals only) for a flat fee of
$19.50 and an hourly charge of $12.50. This means
that cost is a function of the hours the car is rented
plus the flat fee. (a) Write this relationship in
equation form; (b) find the cost if the car is rented
for 3.5 hr; (c) determine how long the car was
rented if the bill came to $119.75; and (d)
determine the domain and range of the function in
this context, if your budget limits you to paying a
maximum of $150 for the rental.
99. Cost of a service call: Paul’s Plumbing charges a
flat fee of $50 per service call plus an hourly rate
of $42.50. This means that cost is a function of the
hours the job takes to complete plus the flat fee.
(a) Write this relationship in equation form;
(b) find the cost of a service call that takes ;
(c) find the number of hours the job took if the
2
1
2
hr
h
8
V1h2 100h
College Algebra—
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charge came to $262.50; and (d) determine the
domain and range of the function in this context, if
your insurance company has agreed to pay for all
charges over $500 for the service call.
100. Predicting tides: The graph
shown approximates the
height of the tides at Fair
Haven, New Brunswick, for a
12-hr period. (a) Is this the
graph of a function? Why?
(b) Approximately what time
did high tide occur? (c) How
high is the tide at 6
P.M.?
(d) What time(s) will the tide be 2.5 m?
101. Predicting tides: The graph
shown approximates the
height of the tides at Apia,
Western Samoa, for a 12-hr
period. (a) Is this the
graph of a function? Why?
(b) Approximately what time
did low tide occur? (c) How
high is the tide at 2
A.M.?
(d) What time(s) will the tide be 0.7 m?
102. Weight of a
fetus: The
growth rate of a
fetus in the
mother’s womb
(by weight in
grams) is
modeled by the
graph shown
here, beginning
with the 25th
week of
gestation. (a) Calculate the average rate of change
(slope of the secant line) between the 25th week
and the 29th week. Is the slope of the secant line
positive or negative? Discuss what the slope means
in this context. (b) Is the fetus gaining weight faster
between the 25th and 29th week, or between the
32nd and 36th week? Compare the slopes of both
secant lines and discuss.
103. Fertility rates:
Over the years,
fertility rates for
women in the
United States
(average number
of children per
woman) have
varied a great
deal, though in
the twenty-first
century they’ve
begun to level out. The graph shown models this
fertility rate for most of the twentieth century.
(a) Calculate the average rate of change from the
years 1920 to 1940. Is the slope of the secant line
positive or negative? Discuss what the slope means
in this context. (b) Calculate the average rate of
change from the year 1940 to 1950. Is the slope of
the secant line positive or negative? Discuss what
the slope means in this context. (c) Was the fertility
rate increasing faster from 1940 to 1950, or from
1980 to 1990? Compare the slope of both secant
lines and comment.
Source: Statistical History of the United States from Colonial
Times to Present
204 CHAPTER 2 Relations, Functions, and Graphs 2-54
Meters
Time
1 A.M. 311953 P.M . 7
2
3
4
1
5
Meters
Time
2 A.M. 4121064 P. M. 8
1.0
0.5
College Algebra—
24
800
1200
1600
2000
26 28
Weight (g)
30 32 34 36 38 40 42
2400
2800
3200
3400
3600
Age (weeks)
(25, 900)
(29, 1100)
(32, 1600)
(36, 2600)
(40, 3200)
3800
Full term
20
1.0
2.0
30 40
Rate (children per woman)
50 60 70 80 90 100 110
(10, 3.4)
(20, 3.2)
(40, 2.2)
(60, 3.6)
(50, 3.0)
(70, 2.4)
(80, 1.8)
(90, 2.0)
10
3.0
4.0
Year (10 → 1910)
EXTENDING THE CONCEPT
104. A father challenges his son to a 400-m race,
depicted in the graph shown here.
a. Who won and what was the approximate
winning time?
Time in seconds
Father:
Son:
Distance in meters
400
300
200
100
0
80706010 20 30 40 50
b. Approximately how many meters behind was
the second place finisher?
c. Estimate the number of seconds the father was
in the lead in this race.
d. How many times during the race were the
father and son tied?
105. Sketch the graph of then discuss how you
could use this graph to obtain the graph of
without computing additional points.
What would the graph of look like?g1x2
x
x
F1x2
x
f1x2 x,
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106. Sketch the graph of then discuss
how you could use this graph to obtain the graph of
without computing additional
points. Determine what the graph of
would look like.
107. If the equation of a function is given, the domain is
implicitly defined by input values that generate real-
g1x2
x
2
4
x
2
4
F1x2
x
2
4
f1x2 x
2
4,
valued outputs. But unless the graph is given or can
be easily sketched, we must attempt to find the range
analytically by solving for x in terms of y. We should
note that sometimes this is an easy task, while at
other times it is virtually impossible and we must rely
on other methods. For the following functions,
determine the implicit domain and find the range by
solving for x in terms of y. a. b. y x
2
3y
x 3
x 2
2-55 Section 2.4 Functions, Function Notation, and the Graph of a Function 205
MAINTAINING YOUR SKILLS
108. (2.2) Which line has a steeper slope, the line
through ( , 3) and (2, 6), or the line through
(0, ) and (9, 4)?
109. (R.6) Compute the sum and product indicated:
a.
b. 12 13
212 132
124
6154 16
4
5
110. (1.5) Solve the equation using the quadratic
formula, then check the result(s) using substitution:
111. (R.4) Factor the following polynomials completely:
a.
b.
c. 8x
3
125
2x
2
13x 24
x
3
3x
2
25x 75
x
2
4x 1 0
1. Sketch the graph of the line
. Plot and label
at least three points.
2. Find the slope of the line
passing through the given
points: and
.
3. In 2002, Data.com lost $2
million. In 2003, they lost $0.5
million. Will the slope of the
line through these points be
positive or negative? Why?
Calculate the slope. Were you
correct? Write the slope as a
unit rate and explain what it
means in this context.
4. Sketch the line passing
through (1, 4) with slope (plot and label at
least two points). Then find the equation of the line
perpendicular to this line through (1, 4).
5. Write the equation for line L
1
shown. Is this the
graph of a function? Discuss why or why not.
6. Write the equation for line L
2
shown. Is this the
graph of a function? Discuss why or why not.
7. For the graph of function h(x) shown, (a) determine
the value of h(2); (b) state the domain; (c) determine
the value of x for which and (d) state the
range.
h1x23;
m
2
3
14, 102
13, 82
4x 3y 12
8. Judging from the appearance of the graph alone,
compare the average rate of change from to
to the rate of change from to
Which rate of change is larger? How is that
demonstrated graphically?
9. Find a linear function that
models the graph of F(p) given.
Explain the slope of the line in
this context, then use your
model to predict the fox
population when the pheasant
population is 20,000.
10. State the domain and range for each function below.
a. b.
c.
55
x
y
5
5
55
x
y
5
5
55
5
x
y
5
x 5.x 4x 2
x 1
MID-CHAPTER CHECK
55
5
L
1
L
2
x
y
5
x
y
5
5
5
5
h(x)
Exercises 5 and 6
Exercises 7 and 8
105678910234
10
Fox population (in 100s)
Pheasant population (1000s)
2
3
1
4
5
6
7
8
9
P
F
F(p)
Exercise 9
College Algebra—
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REINFORCING BASIC CONCEPTS
The Various Forms of a Linear Equation
In a study of mathematics, getting a glimpse of the “big picture” can be an enormous help. Learning mathematics is like
building a skyscraper: The final height of the skyscraper ultimately depends on the strength of the foundation and quality
of the frame supporting each new floor as it is built. Our work with linear functions and their graphs, while having a
number of useful applications, is actually the foundation on which much of your future work will be built. The study of
quadratic and polynomial functions and their applications all have their roots in linear equations. For this reason, it’s
important that you gain a certain fluency with linear functions—even to a point where things come to you effortlessly
and automatically. This level of performance requires a strong desire and a sustained effort. We begin by reviewing the
basic facts a student MUST know to reach this level. MUST is an acronym for m
emorize, understand, synthesize, and
t
each others. Don’t be satisfied until you’ve done all four. Given points (x
1
, y
1
) and (x
2
, y
2
):
Forms and Formulas
slope formula point-slope form slope-intercept form standard form
given any two points given slope m and given slope m and also used in linear
on the line any point (x
1
, y
1
) y-intercept (0, b) systems (Chapter 6)
Characteristics of Lines
y-intercept x-intercept increasing decreasing
(0, y)(x, 0)
let let line slants upward line slants downward
solve for y solve for x from left to right from left to right
Practice for Speed and Accuracy
For the two points given, (a) compute the slope of the line and state whether the line is increasing or decreasing; (b) find
the equation of the line using point-slope form; (c) write the equation in slope-intercept form; (d) write the equation in
standard form; and (e) find the x- and y-intercepts and graph the line.
1. P
1
(0, 5); P
2
(6, 7) 2. P
1
(3, 2); P
2
(0, 9) 3. P
1
(3, 2); P
2
(9, 5)
4. 5. 6. P
1
12, 72; P
2
18, 22P
1
12, 52; P
2
16, 12P
1
15, 42; P
2
13, 22
y 0,x 0,
m 6 0m 7 0
Ax By Cy mx by y
1
m1x x
1
2m
y
2
y
1
x
2
x
1
206 CHAPTER 2 Relations, Functions, and Graphs 2-56
In this section, we’ll consolidate and refine many of the ideas we’ve encountered
related to functions. When functions and graphs are applied as real-world models, we
create a numeric and visual representation that enables an informed response to ques-
tions involving maximum efficiency, positive returns, increasing costs, and other rela-
tionships that can have a great impact on our lives.
A. Graphs and Symmetry
While the domain and range of a function will remain dominant themes in our study,
for the moment we turn our attention to other characteristics of a function’s graph. We
begin with the concept of symmetry.
2.5 Analyzing the Graph of a Function
Learning Objectives
In Section 2.5 you will learn how to:
A. Determine whether a
function is even, odd,
or neither
B. Determine intervals
where a function is
positive or negative
C. Determine where a
function is increasing
or decreasing
D. Identify the maximum
and minimum values
of a function
E. Develop a formula to
calculate rates of
change for any function
College Algebra—
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1010
5
5
x
h(x)
y
(8, 4)
(1, 1)
(0, 0)
(8, 4)
(1, 1)
Figure 2.40
2-57 Section 2.5 Analyzing the Graph of a Function 207
Symmetry with Respect to the y-Axis
Consider the graph of shown in
Figure 2.38, where the portion of the graph to the left
of the y-axis appears to be a mirror image of the portion
to the right. A function is symmetric to the y-axis if,
given any point (x, y) on the graph, the point is
also on the graph. We note that is on the
graph, as is and that is an x-intercept
of the graph, as is (2, 0). Functions that are symmetric
to the y-axis are also known as even functions and in
general we have:
Even Functions: y-Axis Symmetry
A function f is an even function if and only if, for each point (x, y) on the graph of f,
the point ( , y) is also on the graph. In function notation
Symmetry can be a great help in graphing new functions, enabling us to plot fewer
points, and to complete the graph using properties of symmetry.
EXAMPLE 1
Graphing an Even Function Using Symmetry
a. The function g(x) in Figure 2.39 is known to be even. Draw the complete graph
(only the left half is shown).
b. Show that is an even function using
the arbitrary value [show ],
then sketch the complete graph using h(0),
h(1), h(8), and y-axis symmetry.
Solution
a. To complete the graph of g (see Figure 2.39)
use the points ( , 1), ( ), ( , 2), and
y-axis symmetry to find additional points. The
corresponding ordered pairs are (4, 1),
(2, ), and (1, 2), which we use to help draw
a “mirror image” of the partial graph given.
b. To prove that is an even function,
we must show for any
constant k. After writing as , we have:
first step of proof
evaluate and h(k)
radical form
✓ result:
Using , , and with
y-axis symmetry produces the graph shown
in Figure 2.40.
Now try Exercises 7 through 12
h182 4h112 1h102 0
1k2
2
k
2
2
3
k
2
2
3
k
2
2
3
1k2
2
2
3
1k2
2
h 1
k2
31k2
2
4
1
3
31k2
2
4
1
3
h1k2 h1k2
3x
2
4
1
3
x
2
3
h1k2 h1k2
h1x2 x
2
3
3
12, 34
h1k2 h1k2x k
h1x2 x
2
3
f1x2 f1x2
x
12, 0211, 32,
11, 32
1x, y2
(1, 3)
(2, 0)
(2.2, ~4)
(1, 3)
55
5
5
x
y
f(x) x
4
4x
2
(2, 0)
(2.2, ~4)
f1x2 x
4
4x
2
Figure 2.38
55
5
5
x
g(x)
y
(4, 1)
(1, 2)
(1, 2)
(4, 1)
(2, 3) (2, 3)
Figure 2.39
WORTHY OF NOTE
The proof can also be
demonstrated by writing
as , and you are asked to
complete this proof in
Exercise 82.
A
x
1
3
B
2
x
2
3
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208 CHAPTER 2 Relations, Functions, and Graphs 2-58
Symmetry with Respect to the Origin
Another common form of symmetry is known as symmetry to the origin.As the name
implies, the graph is somehow “centered” at (0, 0). This form of symmetry is easy to
see for closed figures with their center at (0, 0), like certain polygons, circles, and
ellipses (these will exhibit both y-axis symmetry and symmetry to the origin). Note
the relation graphed in Figure 2.41 contains the points ( , 3) and (3, ), along with
( ) and (1, 4). But the function f(x) in Figure 2.42 also contains these points and
is, in the same sense, symmetric to the origin (the paired points are on opposite sides
of the x- and y-axes, and a like distance from the origin).
1, 4
33
WORTHY OF NOTE
While the graph of an even
function may or may not
include the point (0, 0), the
graph of an odd function will
always contain this point.
55
5
5
x
y
(3, 3)
(1, 4)
(1, 4)
(3, 3)
55
5
5
x
y
(3, 3)
(1, 4)
(1, 4)
f(x)
(3, 3)
Figure 2.41 Figure 2.42
Figure 2.43 Figure 2.44
Functions symmetric to the origin are known as odd functions and in general we have:
Odd Functions: Symmetry about the Origin
A function f is an odd function if and only if, for each point (x, y) on the graph of f,
the point ( ) is also on the graph. In function notation
EXAMPLE 2
Graphing an Odd Function Using Symmetry
a. In Figure 2.43, the function g(x) given is known to be odd. Draw the complete
graph (only the left half is shown).
b. Show that is an odd function using the arbitrary value
3show , , h(0), and
odd symmetry.
Solution
a. To complete the graph of g, use the points ( , 3), ( , 0), and ( , 2) and
odd symmetry to find additional points. The corresponding ordered pairs are
(6, ), (4, 0), and (2, ), which we use to help draw a “mirror image” of
the partial graph given (see Figure 2.43).
23
246
h112h1x2h1x24, then sketch the graph using h122
x kh1x2 x
3
4x
f1x2f1x2
x, y
1010
10
10
x
y
(2, 2)
(4, 0)
g(x)
(4, 0)
(6, 3)
(2, 2)
(6, 3)
55
5
5
x
y
(1, 3)
(1, 3)
(2, 0) (2, 0)
(0, 0)
h(x)
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2-59 Section 2.5 Analyzing the Graph of a Function 209
b. To prove that is an odd function, we must show that
✓
Using , and with symmetry about the origin
produces the graph shown in Figure 2.44.
Now try Exercises 13 through 24
B. Intervals Where a Function Is Positive or Negative
Consider the graph of shown in Figure 2.45, which has x-intercepts at
( , 0) and (2, 0). Since x-intercepts have the form (x, 0) they are also called the zeroes
of the function (the x-input causes an output of 0). Just as zero on the number line sep-
arates negative numbers from positive numbers, the zeroes of a function that crosses
the x-axis separate x-intervals where a function is negative from x-intervals where the
function is positive. Noting that outputs (y-values) are positive in Quadrants I and II,
in intervals where its graph is above the x-axis. Conversely, in
x-intervals where its graph is below the x-axis. To illustrate, compare the graph of f in
Figure 2.45, with that of g in Figure 2.46.
f1x26 0f1x27 0
2
f1x2 x
2
4
h102 0h122 0, h112 3
k
3
4k k
3
4k
1k2
3
41k2 3k
3
4k4
h1k2 h1k2
h1k2h1k2.
h1x2 x
3
4x
Figure 2.45
Figure 2.46
A. You’ve just learned how
to determine whether a
function is even, odd, or
neither
(0, 4)
(2, 0)
55
5
5
x
y
f(x) x
2
4
(2, 0)
(4, 0)
53
5
5
x
y
g(x) (x 4)
2
The graph of f is a parabola, with x-intercepts of ( , 0) and (2, 0). Using our pre-
vious observations, we note for for
. The graph of g is also a parabola, but is entirely above or on the x-axis,
showing for . The difference is that zeroes coming from factors of the
form ( ) (with degree 1) allow the graph to cross the x-axis. The zeroes of f came
from . Zeroes that come from factors of the form (with
degree 2) cause the graph to “bounce” off the x-axis since all outputs must be non-
negative. The zero of g came from .
EXAMPLE 3
Solving an Inequality Using a Graph
Use the graph of given to solve the inequalities
a.
b. g1x26 0
g1x2 0
g1x2 x
3
2x
2
4x 8
1x 42
2
0
1x r2
2
1x 221x 22 0
x r
x g1x2 0
x 12, 22
x 1q, 24 ´ 32, q2 and f1x26 0f 1x2 0
2
WORTHY OF NOTE
These observations form the
basis for studying polynomi-
als of higher degree, where
we extend the idea to factors
of the form in a study
of roots of multiplicity
(also see the Calculator
Exploration and Discovery
feature in this chapter).
1x r2
n
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